A007918 -id:A007918 - cp's OEIS frontend

A359300 a(n) = (distance from n to nearest prime >= n) - (distance from n to nearest prime <= n).

0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 0, 2, 0, -2, 0, 0, 0, 2, 0, -2, 0, 4, 2, 0, -2, -4, 0, 0, 0, 4, 2, 0, -2, -4, 0, 2, 0, -2, 0, 0, 0, 2, 0, -2, 0, 4, 2, 0, -2, -4, 0, 4, 2, 0, -2, -4, 0, 0, 0, 4, 2, 0, -2, -4, 0, 2, 0, -2, 0, 0, 0, 4, 2, 0, -2, -4, 0, 2, 0

Offset: 2

Clark Kimberling, Jan 01 2023

sign

			a(8) = 2 because (11-8) - (8-7) = 2.

Cf. A000040, A007917, A007918, A008578, A359298, A359299.

f:= n -> nextprime(n-1) + prevprime(n+1) - 2*n:
map(f, [$2..100]); # Robert Israel, Jan 03 2023

u[n_] := If[PrimeQ[n], n, NextPrime[n]];
v[n_] := If[PrimeQ[n], n, NextPrime[n, -1]];
Table[u[n] - n - (n - v[n]), {n, 2, 350}]

a(n) = A007917(n) + A007918(n) - 2*n. - Bernard Schott, Jan 01 2023

A378252 Least prime power > 2^n.

Original entry on oeis.org

2, 3, 5, 9, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659, 4294967311, 8589934609

Offset: 0

Gus Wiseman, Nov 30 2024

nonn

Prime powers are listed by A246655.

Conjecture: All terms except 9 are prime. Hence this is the same as A014210 after 9. Confirmed up to n = 1000.

Subtracting 2^n appears to give A013597 except at term 3.
For prime we have A014210.
For previous we have A014234.
For perfect power we have A357751.
For squarefree we have A372683.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, diffs A375708 and A375735.
A031218 gives the greatest prime power <= n.
A244508 counts prime powers between powers of 2.
Prime powers between primes are counted by A080101 and A366833.
Cf. A007918, A037035, A053707, A059305, A065514, A069584, A151800, A304521, A377051, A377282, A377283, A377289, A377432, A378249.

Table[NestWhile[#+1&,2^n+1,!PrimePowerQ[#]&],{n,0,20}]

a(n) = my(x=2^n+1); while (!isprimepower(x), x++); x; \\ Michel Marcus, Dec 03 2024

from itertools import count
from sympy import primefactors
def A378252(n): return next(i for i in count(1+(1<Chai Wah Wu, Dec 02 2024

A049852 Concatenate "n" and "nextprime(n)".

Original entry on oeis.org

12, 23, 35, 45, 57, 67, 711, 811, 911, 1011, 1113, 1213, 1317, 1417, 1517, 1617, 1719, 1819, 1923, 2023, 2123, 2223, 2329, 2429, 2529, 2629, 2729, 2829, 2931, 3031, 3137, 3237, 3337, 3437, 3537, 3637, 3741, 3841, 3941, 4041, 4143, 4243

Offset: 1

N. J. A. Sloane

From Petros Hadjicostas, Nov 20 2019: (Start)
Version 1 of the "next prime" function is A007918: smallest prime >= n. PARI/GP's nextprime() is version 1.
Maple's nextprime() is the version 2 that appears in A151800: smallest prime > n.  We use version 2 here. (End)

			From _Petros Hadjicostas_, Nov 20 2019: (Start)
a(1) = 12 because nextprime(1) = 2.
a(2) = 23 because nextprime(2) = 3.
a(3) = 35 because nextprime(3) = 5.
a(4) = 45 because nextprime(4) = 5.
...
a(10) = 1011 because nextprime(10) = 11.
a(11) = 1113 because nextprime(11) = 13.
... (End)

Cf. A007918 (version 1 of nextprime), A151800 (version 2 of nextprime).

a:= n-> parse(cat(n, nextprime(n))):
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 20 2019

a(n) = eval(concat(Str(n), Str(nextprime(n+1)))); \\ Michel Marcus, Jan 01 2017

A060847 Difference between a nontrivial prime power (A246547) and the previous prime.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 1, 2, 3, 2, 8, 12, 1, 2, 2, 5, 6, 6, 2, 3, 6, 6, 2, 2, 8, 3, 4, 2, 12, 2, 9, 8, 18, 2, 2, 6, 4, 12, 2, 3, 6, 4, 2, 6, 12, 8, 2, 6, 2, 1, 6, 8, 2, 2, 14, 4, 6, 2, 6, 2, 3, 20, 2, 12, 2, 2, 8, 14, 10, 18, 8, 6, 2, 2, 2, 12, 12, 19, 2, 6, 6, 20, 2, 2, 2, 8, 8, 2, 2, 8, 20, 12, 15, 2, 4

Offset: 1

Labos Elemer, May 03 2001

nonn

a(n)=1 only for some powers of 2.

			78125=5^7 follows 78121, the difference is 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A246547.

N:= 10^5: # to consider prime powers <= N
P:= select(isprime,[2,seq(i,i=3..floor(sqrt(N)),2)]):
PP:= sort([seq(seq(p^k,k=2..ilog[p](N)),p=P)]):
map(t -> t - prevprime(t), PP); # Robert Israel, Nov 13 2024

from sympy import primepi, integer_nthroot, prevprime
def A060847(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return (a:=bisection(f,n,n))-prevprime(a) # Chai Wah Wu, Sep 13 2024

a(n) = A246547(n)-prevprime(A246547(n)) = A246547(n)-A049711(A246547(n)).

A060848 Difference between a nontrivial prime power (A025475) and the next prime.

Original entry on oeis.org

1, 3, 2, 1, 4, 2, 5, 4, 3, 2, 6, 2, 3, 4, 8, 1, 4, 4, 6, 9, 12, 6, 4, 12, 6, 7, 30, 4, 12, 12, 5, 16, 6, 4, 10, 10, 12, 10, 6, 3, 4, 6, 10, 4, 6, 2, 4, 10, 6, 17, 4, 10, 4, 18, 6, 30, 12, 12, 4, 10, 27, 4, 6, 4, 12, 4, 28, 6, 2, 10, 4, 4, 10, 12, 18, 10, 10, 3, 12, 4, 12, 6, 10, 10, 18, 10, 12

Offset: 1

Labos Elemer, May 03 2001

nonn

a(n)=1 only for some powers of 2 corresponding to Fermat primes > 3. - Edited by Robert Israel, Jun 03 2021

			78125=5^7 is followed by 78137, the difference is 12.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A025475, A007918, A013632.

N:= 10^5: # for prime powers <= N
S:= {}:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  S:= S union {seq(p^i,i=2..floor(log[p](N)))}
od:
map(t -> nextprime(t)-t, sort(convert(S,list))); # Robert Israel, Jun 03 2021

NextPrime[#]-#&/@Select[Range[100000],PrimePowerQ[#]&&!PrimeQ[#]&] (* Harvey P. Dale, Oct 19 2022 *)

a(n) = nextprime(A025475(n)) - A025475(n) = A013632(A025475(n)).

A060849 Difference between 2 consecutive primes between which a nontrivial power of prime is found.

Original entry on oeis.org

2, 4, 4, 4, 6, 6, 6, 6, 6, 4, 14, 14, 4, 6, 10, 6, 10, 10, 8, 12, 18, 12, 6, 14, 14, 10, 34, 6, 24, 14, 14, 24, 24, 6, 12, 16, 16, 22, 8, 6, 10, 10, 12, 10, 18, 10, 6, 16, 8, 18, 10, 18, 6, 20, 20, 34, 18, 14, 10, 12, 30, 24, 8, 16, 14, 6, 36, 20, 12, 28, 12, 10, 12, 14, 20, 22, 22

Offset: 1

Labos Elemer, May 03 2001

nonn

			59049=3^10 is between 59029 and 59051, so the corresponding term is 59051-59029=22.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A246547, A007917, A007918, A013632, A013633, A049711.

ispp(n) = isprimepower(n) >= 2; \\ A246547
lista(nn) = {for (n=1, nn, if (ispp(n), print1(nextprime(n) - precprime(n), ", ")););} \\ Michel Marcus, Mar 23 2020

a(n) = nextprime(A246547(n)) - prevprime(A246547(n)) = A013633(A246547(n)). [corrected by Michel Marcus, Mar 23 2020]

A079476 First prime greater than or equal to phi(n^2).

Original entry on oeis.org

2, 2, 7, 11, 23, 13, 43, 37, 59, 41, 113, 53, 157, 89, 127, 131, 277, 109, 347, 163, 257, 223, 509, 193, 503, 313, 487, 337, 821, 241, 937, 521, 661, 547, 853, 433, 1361, 691, 937, 641, 1657, 509, 1811, 881, 1087, 1013, 2179, 769, 2063, 1009, 1637, 1249, 2767

Offset: 1

Jon Perry, Jan 15 2003

The sequence generally goes up,down,up,down...

			phi(3^2)=phi(9)=6, therefore a(3)=7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002618, A007918.

map(t -> nextprime(numtheory:-phi(t^2)-1), [$1..100]); # Robert Israel, Oct 25 2017

Array[If[PrimeQ@ #, #, NextPrime@ #] &@ EulerPhi[#^2] &, 53] (* Michael De Vlieger, Oct 25 2017 *)

for (n=1,100, print1(nextprime(eulerphi(n^2))","))

a(n) = A007918(A002618(n)). - Robert Israel, Oct 25 2017

Corrected by Robert Israel, Oct 25 2017

A084571 Let a(1)=1; for n>1, a(n)=nextprime((3/2)*a(n-1)).

Original entry on oeis.org

1, 2, 3, 5, 11, 17, 29, 47, 71, 107, 163, 251, 379, 569, 857, 1289, 1949, 2927, 4391, 6599, 9901, 14867, 22303, 33457, 50207, 75323, 112997, 169501, 254257, 381389, 572087, 858149, 1287233, 1930879, 2896319, 4344479, 6516739, 9775111, 14662727

Offset: 1

Paul D. Hanna, May 30 2003

nonn

The definition refers to the nextprime() function in A007918.

Cf. A084572.

Join[{1,2},NestList[NextPrime[(3/2)*#]&,3,36]] (* Jayanta Basu, May 26 2013 *)

a(n)=if(n<2,1,nextprime((3/2)*a(n-1))); for(n=1,50,print1(a(n),","))

A084748 Primes of the form n! + p where p is the smallest prime > n.

Original entry on oeis.org

3, 5, 11, 29, 127, 727, 5051, 3628811, 8683317618811886495518194401280000037, 2658271574788448768043625811014615890319638528000000047, 12413915592536072670862289047373375038521486354677760000000053

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 16 2003

nonn

Let a term k! + p = prime(r) then prime(r) -prime(r-1) >=p-1.

			727 = 6! + 7 belongs to this sequence but 8! +11 is composite and is not a member.

Cf. A084749.

a(n) = A084749(n)!+A007918(A084749(n)+1). - David Wasserman, Jan 04 2005

More terms from David Wasserman, Jan 04 2005

A097521 Difference between F(n) = 2^(2^n)+1 and nextprime(F(n)) where F(n) is the n-th Fermat Number.

Original entry on oeis.org

2, 2, 2, 6, 2, 14, 12, 50, 296, 74, 642, 980, 1760, 896, 2774, 118112, 44060, 5850

Offset: 0

Cino Hilliard, Aug 27 2004

In the Name, nextprime means A151800, not A007918. - Jeppe Stig Nielsen, Nov 18 2019

a(n) = A129786(n)-1 except in the rare case that F(n) is a prime. - Jeppe Stig Nielsen, Nov 18 2019

			F(5) = 4294967297. Nextprime(F(5)) = 4294967311.
4294967311 - 4294967297 = 14 the 6th entry in the table.

Cf. A000215, A013632, A151800, A129786.

for(n=0,+oo,print1(nextprime(2^(2^n)+2)-(2^(2^n)+1),", ")) \\ Jeppe Stig Nielsen, Nov 18 2019

a(n) = A013632(A000215(n)). - Michel Marcus, Nov 18 2019

More terms with the help of A129786 from Jeppe Stig Nielsen, Nov 18 2019

cp's OEIS Frontend

A359300 a(n) = (distance from n to nearest prime >= n) - (distance from n to nearest prime <= n).

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Maple

Mathematica

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A378252 Least prime power > 2^n.

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Mathematica

PARI

Python

A049852 Concatenate "n" and "nextprime(n)".

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Maple

PARI

A060847 Difference between a nontrivial prime power (A246547) and the previous prime.

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Maple

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A060848 Difference between a nontrivial prime power (A025475) and the next prime.

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Maple

Mathematica

Formula

A060849 Difference between 2 consecutive primes between which a nontrivial power of prime is found.

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PARI

Formula

A079476 First prime greater than or equal to phi(n^2).

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Maple

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A084571 Let a(1)=1; for n>1, a(n)=nextprime((3/2)*a(n-1)).